Algebraic & Geometric Topology

An algorithm to determine the Heegaard genus of simple $3$–manifolds with nonempty boundary

Marc Lackenby

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We provide an algorithm to determine the Heegaard genus of simple 3–manifolds with nonempty boundary. More generally, we supply an algorithm to determine (up to ambient isotopy) all the Heegaard splittings of any given genus for the manifold. As a consequence, the tunnel number of a hyperbolic link is algorithmically computable. Our techniques rely on Rubinstein’s work on almost normal surfaces, and also on a new structure called a partially flat angled ideal triangulation.

Article information

Algebr. Geom. Topol., Volume 8, Number 2 (2008), 911-934.

Received: 7 February 2008
Revised: 1 May 2008
Accepted: 2 May 2008
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Heegaard algorithm 3-manifold


Lackenby, Marc. An algorithm to determine the Heegaard genus of simple $3$–manifolds with nonempty boundary. Algebr. Geom. Topol. 8 (2008), no. 2, 911--934. doi:10.2140/agt.2008.8.911.

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  • F Bonahon, Cobordism of automorphisms of surfaces, Ann. Sci. École Norm. Sup. $(4)$ 16 (1983) 237–270
  • D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67–80
  • W Jaco, J Rubinstein, Layered-triangulations of $3$–manifolds
  • W Jaco, J L Tollefson, Algorithms for the complete decomposition of a closed $3$–manifold, Illinois J. Math. 39 (1995) 358–406
  • S Kojima, Polyhedral decomposition of hyperbolic $3$–manifolds with totally geodesic boundary, from: “Aspects of low-dimensional manifolds”, Adv. Stud. Pure Math. 20, Kinokuniya, Tokyo (1992) 93–112
  • M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243–282
  • T Li, Heegaard surfaces and measured laminations. II. Non-Haken $3$–manifolds, J. Amer. Math. Soc. 19 (2006) 625–657
  • T Li, Heegaard surfaces and measured laminations. I. The Waldhausen conjecture, Invent. Math. 167 (2007) 135–177
  • S Matveev, Algorithmic topology and classification of $3$–manifolds, Algorithms and Computation in Math. 9, Springer, Berlin (2003)
  • J W Morgan, On Thurston's uniformization theorem for three-dimensional manifolds, from: “The Smith conjecture (New York, 1979)”, Pure Appl. Math. 112, Academic Press, Orlando, FL (1984) 37–125
  • C Petronio, J R Weeks, Partially flat ideal triangulations of cusped hyperbolic $3$–manifolds, Osaka J. Math. 37 (2000) 453–466
  • J H Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for $3$–dimensional manifolds, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 1–20
  • M Scharlemann, J Schultens, The tunnel number of the sum of $n$ knots is at least $n$, Topology 38 (1999) 265–270
  • M Scharlemann, A Thompson, Thin position for $3$–manifolds, from: “Geometric topology (Haifa, 1992)”, Contemp. Math. 164, Amer. Math. Soc. (1994) 231–238
  • J Schultens, The classification of Heegaard splittings for (compact orientable surface)$\,\times\, S\sp 1$, Proc. London Math. Soc. $(3)$ 67 (1993) 425–448
  • M Stocking, Almost normal surfaces in $3$–manifolds, Trans. Amer. Math. Soc. 352 (2000) 171–207
  • J L Tollefson, Involutions of sufficiently large $3$–manifolds, Topology 20 (1981) 323–352